The modified Langevin description for probes in a nonlinear medium

Nonlinear Langevin functionals for a driven probe

When a probe particle immersed in a fluid with nonlinear interactions is subject to strong driving, the cumulants of the stochastic force acting on the probe are nonlinear functionals of the driving protocol. We present a Volterra series for these nonlinear functionals by applying nonlinear response theory in a path integral formalism, where the emerging kernels are shown to be expressed in terms of connected equilibrium correlation functions. The first cumulant is the mean force, the second cumulant characterizes the non-equilibrium force fluctuations (noise), and higher order cumulants quantify non-Gaussian fluctuations. We discuss the interpretation of this formalism in relation to Langevin dynamics. We highlight two example scenarios of this formalism. (i) For a particle driven with the prescribed trajectory, the formalism yields the non-equilibrium statistics of the interaction force with the fluid. (ii) For a particle confined in a moving trapping potential, the formalism yields the non-equilibrium statistics of the trapping force. In simulations of a model of nonlinearly interacting Brownian particles, we find that nonlinear phenomena, such as shear-thinning and oscillating noise covariance, appear in third- or second-order response, respectively.

Memory-induced alignment of colloidal dumbbells

When a colloidal probe is forced through a viscoelastic fluid which is characterized by a long stress-relaxation time, the fluid is excited out of equilibrium. This is leading to a number of interesting effects including a non-trivial recoil of the probe when the driving force is removed. Here, we experimentally and theoretically investigate the transient recoil dynamics of non-spherical particles, i.e., colloidal dumbbells. In addition to a translational recoil of the dumbbells, we also find a pronounced angular reorientation which results from the relaxation of the surrounding fluid. Our findings are in good agreement with a Langevin description based on the symmetries of a director (dumbbell) as well as a microscopic bath-rod model. Remarkably, we find an instability with amplified fluctuations when the dumbbell is oriented perpendicular to the direction of driving. Our results demonstrate the complex behavior of non-spherical objects within a relaxing environment which are of immediate interest for the motion of externally but also self-driven asymmetric objects in viscoelastic fluids.

Anomalous Transport of Tracers in Active Baths

We derive the long-time dynamics of a tracer immersed in a one-dimensional active bath. In contrast to previous studies, we find that the damping and noise correlations possess long-time tails with exponents that depend on the tracer symmetry. For generic tracers, shape asymmetry induces ratchet effects that alter fluctuations and lead to superdiffusion and friction that grows with time when the tracer is dragged at a constant speed. In the singular limit of a completely symmetric tracer, we recover normal diffusion and finite friction. Furthermore, for small symmetric tracers, the active contribution to the friction becomes negative: active particles enhance motion rather than oppose it. These results show that, in low-dimensional systems, the motion of a passive tracer in an active bath cannot be modeled as a persistent random walker with a finite correlation time.

Mori-Zwanzig projection operator formalism: Particle-based coarse-grained dynamics of open classical systems far from equilibrium

We present a generalized Langevin equation (GLE) of motion that governs exactly the time evolution of phase-space observables in finite open systems described by classical Hamiltonians with explicitly time-dependent potentials. This formalism is based on the Mori-Zwanzig projection operator (PO) method with a time-independent Zwanzig PO within a Heisenberg (Lagrangian) picture and reduced description of Hamiltonian systems in terms of canonical relevant and irrelevant coordinates. We demonstrate that, similarly to closed systems, GLE dynamics in Hamiltonian systems in the presence of time-dependent potentials is determined by conservative, dissipative memory, and projected force fields, and that the memory functions relate to the projected force, which is a two-time process, in a way that is reminiscent of the equilibrium second fluctuation-dissipation relation. We further show that, in the most general case, the memory kernel depends on the relevant momentum gradients of the (Boltzmann) entropy of the irrelevant subsystem. Using two Zwanzig operators which are, respectively, functionals of the canonical and generalized canonical probability densities, we then derive what we call canonical and generalized canonical GLEs. Further, we can formulate the particle-based, coarse-grained (CG) GLE dynamics by transitioning to Jacobi coordinates which corresponds to a particle set partitioning of the Hamiltonian system. The obtained canonical CG GLE of motion for the relevant momenta is a generalization of the CG equation of motion known for closed systems. Also, using a Markovian approximation of the canonical CG GLE, we can extend the dissipative particle dynamics equation to open systems. A distinctive feature of our extension is a use of explicitly time-dependent frictions, which reflect the changes in the dissipation rate caused by time-dependent coupling to an external bath. Our GLE formalism and workflow constitute a general and viable framework that can be readily used as a starting point to rigorously formulate microscopically informed CG treatments for a variety of phenomena in externally forced systems far from equilibrium.

Two step micro-rheological behavior in a viscoelastic fluid

We perform micro-rheological experiments with a colloidal bead driven through a viscoelastic worm-like micellar fluid and observe two distinctive shear thinning regimes, each of them displaying a Newtonian-like plateau. The shear thinning behavior at larger velocities is in qualitative agreement with macroscopic rheological experiments. The second process, observed at Weissenberg numbers as small as a few percent, appears to have no analog in macro-rheological findings. A simple model introduced earlier captured the observed behavior and implied that the two shear thinning processes correspond to two different length scales in the fluid. This model also reproduces oscillations, which have been observed in this system previously. While the system under macro-shear seems to be near equilibrium for shear rates in the regime of the intermediate Newtonian-like plateau, the one under micro-shear is thus still far from it. The analysis suggests the existence of a length scale of a few micrometres, the nature of which remains elusive.

Fluctuating Motion in an Active Environment

We derive the fluctuation dynamics of a probe in weak coupling with a living medium, modeled as particles undergoing an active Ornstein-Uhlenbeck dynamics. Nondissipative corrections to the fluctuation-dissipation relation are written out explicitly in terms of time correlations in the active medium. A first term changes the inertial mass of the probe as a consequence of the persistence of the active medium. A second correction modifies the friction kernel. The resulting generalized Langevin equation benchmarks the motion induced on probes immersed in active versus passive media. The derivation uses nonequilibrium response theory.

Stochastic simulation of anharmonic dissipation. II. Harmonic bath potentials with quadratic couplings

The workhorse simulating the dissipative dynamics is mainly based on the harmonic bath potentials together with linear system-bath couplings, but a realistic bath always assumes anharmonicity. In this work, we extend the linear dissipation model to include quadratic couplings and suggest a stochastic simulation scheme for the anharmonic dissipation. We show that the non-Gaussian noises induced by the anharmonic bath can be rigorously constructed, and the resulting stochastic Liouville equation has the same form as that for the linear dissipation model. As a preliminary application, we use this stochastic method to investigate the vibration-induced symmetry breaking in two-level electronic systems and find that the characteristic function of the non-Gaussian noises determines the absorption and fluorescence spectra.

The Role of Data in Model Building and Prediction: A Survey Through Examples

The goal of Science is to understand phenomena and systems in order to predict their development and gain control over them. In the scientific process of knowledge elaboration, a crucial role is played by models which, in the language of quantitative sciences, mean abstract mathematical or algorithmical representations. This short review discusses a few key examples from Physics, taken from dynamical systems theory, biophysics, and statistical mechanics, representing three paradigmatic procedures to build models and predictions from available data. In the case of dynamical systems we show how predictions can be obtained in a virtually model-free framework using the methods of analogues, and we briefly discuss other approaches based on machine learning methods. In cases where the complexity of systems is challenging, like in biophysics, we stress the necessity to include part of the empirical knowledge in the models to gain the minimal amount of realism. Finally, we consider many body systems where many (temporal or spatial) scales are at play—and show how to derive from data a dimensional reduction in terms of a Langevin dynamics for their slow components.

Memory-Induced Transition from a Persistent Random Walk to Circular Motion for Achiral Microswimmers

We experimentally study the motion of light-activated colloidal microswimmers in a viscoelastic fluid. We find that, in such a non-Newtonian environment, the active colloids undergo an unexpected transition from enhanced angular diffusion to persistent rotational motion above a critical propulsion speed, despite their spherical shape and stiffness. We observe that, in contrast to chiral asymmetric microswimmers, the resulting circular orbits can spontaneously reverse their sense of rotation and exhibit an angular velocity and a radius of curvature that nonlinearly depend on the propulsion speed. By means of a minimal non-Markovian Langevin model for active Brownian motion, we show that these nonequilibrium effects emerge from the delayed response of the fluid with respect to the self-propulsion of the particle without counterpart in Newtonian fluids.

Unified picture of strong-coupling stochastic thermodynamics and time reversals

Strong-coupling statistical thermodynamics is formulated as the Hamiltonian dynamics of an observed system interacting with another unobserved system (a bath). It is shown that the entropy production functional of stochastic thermodynamics, defined as the log ratio of forward and backward system path probabilities, is in a one-to-one relation with the log ratios of the joint initial conditions of the system and the bath. A version of strong-coupling statistical thermodynamics where the system-bath interaction vanishes at the beginning and at the end of a process is, as is also weak-coupling stochastic thermodynamics, related to the bath initially in equilibrium by itself. The heat is then the change of bath energy over the process, and it is discussed when this heat is a functional of the system history alone. The version of strong-coupling statistical thermodynamics introduced by Seifert and Jarzynski is related to the bath initially in conditional equilibrium with respect to the system. This leads to heat as another functional of the system history which needs to be determined by thermodynamic integration. The log ratio of forward and backward system path probabilities in a stochastic process is finally related to log ratios of the initial conditions of a combined system and bath. It is shown that the entropy production formulas of stochastic processes under a general class of time reversals are given by the differences of bath energies in a larger underlying Hamiltonian system. The paper highlights the centrality of time reversal in stochastic thermodynamics, also in the case of strong coupling.

Oscillating modes of driven colloids in overdamped systems

Microscopic colloidal particles suspended in liquids are a prominent example of an overdamped system where viscous forces dominate over inertial effects. Frequently, colloids are used as sensitive probes, e.g., in biophysical applications from which molecular forces are inferred. The interpretation of such experiments rests on the assumption that, even when the particles are driven, the liquid remains in equilibrium. Here we experimentally demonstrate that this is not valid for particles in viscoelastic fluids. Even at small driving forces, we observe particle oscillations with several tens of seconds. They are attributed to non-equilibrium fluctuations of the fluid being excited by the particle’s motion. The oscillatory dynamics is in quantitative agreement with an overdamped Langevin equation with negative friction-memory term being equivalent to a stochastically driven underdamped oscillator. Such oscillatory modes are expected to widen the use of colloids as model systems but must also be considered in colloidal probe experiments.

The motion of microparticles suspended in liquids is assumed to be dominated by viscous forces. Here, Berner et al. challenge this consensus by observing underdamped particle oscillations in a viscoelastic fluid and attributing it to the non-equilibrium fluctuations of liquid excited by particles.